Question

Draw any obtuse angle bisect it using compass

Answer 1

Given angle
∠
A
B
C
with vertex
B
and two sides
B
A
and
B
C
. It can be acute or obtuse, or right - makes no difference.

Choose any segment of some length
d
and mark point
M
on side
B
A
on a distance
d
from vertex
B
.
Using the same segment of length
d
, mark point
N
on side
B
C
on distance
d
from vertex
B
.
Red arc on a picture represents this process, its ends are
M
and
N
.

We can say now that
B
M
≅
B
N
.

Choose a radius sufficiently large (greater than half the distance between points
M
and
N
) and draw two circles with centers at points
M
and
N
of this radius. These two circles intersect in two points,
P
and
Q
. See two small arcs intersecting on a picture, their intersection is point
P
.

Chose any of these intersection points, say
P
, and connect it with vertex
B
. This is a bisector of an angle
∠
A
B
C
.

Proof

Compare triangles
Δ
B
M
P
and
Δ
B
N
P
.
1. They share side
B
P

2.
B
M
≅
B
N
by construction, since we used the same length
d
to mark both points
M
and
N

3.
M
P
≅
N
P
by construction, since we used the same radius of two intersecting circles with centers at points
M
and
N
.
Therefore, triangles
Δ
B
M
P
and
Δ
B
N
P
are congruent by three sides:
Δ
B
M
P
≅
Δ
B
N
P

As a consequence of congruence of these triangles, corresponding angles have the same measure.
Angles
∠
M
B
P
and
∠
N
B
P
lie across congruent sides
M
P
and
N
P
.
Therefore, these angles are congruent:
∠
M
B
P
≅
∠
N
B
P
,
that is
B
P
is a bisector of angle
∠
M
B
P
(which is the same as angle
∠
A
B
C
).

Answer 2

draw a straight line